According to [1]}, with probability at least \(1-\delta /3\) , we have
\(\begin{split}&\sup _{\tilde{h}_{\theta _f +\Delta \theta _j}- h_{\theta _f}\in \mathcal {F}} \left\lbrace \left\Vert \tilde{h}_{\theta _f +\Delta \theta _j}(x^{(j)}) - h_{\theta _f}(x^{(j)}) - \left(f_{0,-j}(x^{(j)}) - h_{\theta _f}(x^{(j)})\right)\right\Vert  - \Vert \tilde{h}_{\theta _f +\Delta \theta _j}(x) - f_{0,-j}(x^{(j)})\Vert _n \right\rbrace  \\\le & 2\hat{\mathcal {R}}_S(\mathcal {F}) + \sqrt{\frac{\log (3/\delta )}{2n}}.\end{split}\) 
